Nonlinear generation of sound and mean pressure drop in shock–shear interaction

Abstract

Interactions of shear fluctuations with shock waves are ubiquitous in high-speed flow applications from scramjet propulsion to cosmic events like supernova explosions. They also serve as fundamental building blocks for the study of shock–turbulence interaction. In this work, we study the nonlinear effects in pressure arising due to the interaction of a normal shock with a two-dimensional shear wave. It employs the weakly nonlinear framework (WNLF) developed recently for vorticity amplification by Thakare et al. [“A weakly nonlinear framework to study shock–vorticity interaction,” J. Fluid Mech. 933, A48 (2022)]. The analysis includes the effect of intermodal interactions that is neglected in the widely used linear interaction analysis (LIA) of shock–turbulence interaction. It is found that the deformation of the shock wave and the fluctuation mass flux normal to the shock contribute to the dominant physical mechanisms responsible for the observed nonlinearities. Interestingly, the WNLF predicts a drop in mean pressure behind the shock due to a second-order intermodal interaction, which is consistent with the well-established results by Lele [“Shock-jump relations in a turbulent flow,” Phys. Fluids A 4, 2900–2905 (1992)] at low Mach numbers and brings out additional effects of shock deformation that are important at higher Mach numbers. We extend the WNLF to three-dimensional interaction of homogeneous isotropic turbulence with a normal shock. Comparison with existing direct numerical simulation data shows good agreement for low turbulent Mach numbers, which is a significant improvement over the prediction capability of LIA. We also compute the dilatation fields from WNLF and use them to distinguish between the acoustic and non-acoustic components of the second-order pressure fluctuations generated by the shock wave.